Getting the best use of an hourly in home care caregiver.

You wife must know that it's either her sanity or MIL's demands fulfilled.
This is what worked for me when my Mother lived with me: I got her up in the morning, had a nice breakfast with her, got her showered and dressed, then the caregiver would arrive. I had usually been up several times in the night with her, so by 10 am I was ready for some time to myself.
Sometimes I would work in my office, some days I just cried from exhaustion, sometimes I would leave the house. It is not the same as really being free, but at least it relieves one for a few precious hours of the annoying neediness. We had another caregiver who would take Mother out of the house on a daily excursion to the beach or park, or one of the senior centers so that I could be in my own home in peace for a few hours, cleaning or doing yard work. Oh yea, cleaning my house was my idea of a vacation when she lived here. Obviously, I am not your best example of a selfless, doting caregiver, but I'm sure you will get answers from them, too.
I gave my Mother very good care, but did not take care of myself very well, and caregivers must be healthy and rested. Put your Own oxygen mask on first.
I visit her several times a week and do a lot of things for her in the care home where she is on hospice. Encourage your wife to save herself NOW. Bless you, good husband.

What types of things is your wife doing for MIL? Going for a walk? Fixing her lunch? Helping with laundry, making her bed? Caregivers can do all that. Often they just watch TV together and chat part of the time. Is there some reason MIL can't be alone while your wife go out in afternoon? Or is MIL just being clingy? Set up some boundaries and perhaps regular times your wife is gone so MIL gets use to being alone and finding her own way. For weekends and long vacations you certainly may want the caregiver to stay over but most will do whatever normal household and personal care you think MIL needs. Talk to an agency, they can give you an idea of what their caregivers do for other clients.